An inversion sequence of length $n$ is a sequence of integers $e=e_1\cdots e_n$ which satisfies ... more An inversion sequence of length $n$ is a sequence of integers $e=e_1\cdots e_n$ which satisfies for each $i\in[n]=\{1,2,\ldots,n\}$ the inequality $0\le e_i < i$. For a set of patterns $P$, we let $\mathbf{I}_n(P)$ denote the set of inversion sequences of length $n$ that avoid all the patterns from~$P$. We say that two sets of patterns $P$ and $Q$ are I-Wilf-equivalent if $|\mathbf{I}_n(P)|=|\mathbf{I}_n(Q)|$ for every~$n$. In this paper, we show that the number of I-Wilf-equivalence classes among triples of length-3 patterns is $137$, $138$ or~$139$. In particular, to show that this number is exactly $137$, it remains to prove $\{101,102,110\}\stackrel{\mathbf{I}}{\sim}\{021,100,101\}$ and $\{100,110,201\}\stackrel{\mathbf{I}}{\sim}\{100,120,210\}$.
Discrete Mathematics & Theoretical Computer Science
Combinatorics We consider the sum u of differences between adjacent letters of a word of n letter... more Combinatorics We consider the sum u of differences between adjacent letters of a word of n letters, chosen uniformly at random from a given alphabet. This paper obtains the enumerating generating function for the number of such words with respect to the sum u, as well as explicit formulas for the mean and variance of u.
Discrete Mathematics & Theoretical Computer Science
Let asc and desc denote respectively the statistics recording the number of ascents or descents i... more Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and ...
In this paper we introduce the concept of s-Fuss-Catalan words. This new family of words generali... more In this paper we introduce the concept of s-Fuss-Catalan words. This new family of words generalizes the Catalan words (taking s = 1), which are a particular case of growth-restricted words. Here we enumerate the polyominoes or bargraphs associated with the s-Fuss-Catalan words according to the semiperimeter and area statistics. Additionally, we obtain combinatorial formulas to count the s-Fuss-Catalan bargraphs according of these statistics.
We introduce a decomposition method for column-convex polyominoes and enumerate them in terms of ... more We introduce a decomposition method for column-convex polyominoes and enumerate them in terms of two statistics: the number of internal vertices and the number of corners in the boundary. We first find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of interior vertices. In particular, we show that the average number of interior vertices over all column-convex polyominoes of perimeter 2 n is asymptotic to $$\alpha _o n^{3/2}$$ α o n 3 / 2 where $$\alpha _o\approx 0.57895563\ldots .$$ α o ≈ 0.57895563 … . We also find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of corners in the boundary. In particular, we show that the average number of corners over all column-convex polyominoes of perimeter 2 n is asymptotic to $$\alpha _1n$$ α 1 n where $$\alpha _1\approx 1.17157287\ldots .$$ α 1 ≈ 1.17157287 … .
In this paper, we study the Kantorovich variant of Chlodowsky Szász operators induced by Boas-Buc... more In this paper, we study the Kantorovich variant of Chlodowsky Szász operators induced by Boas-Buck type polynomials and prove a Korovkin type theorem. Also, we present the Voronovskaya type theorem and Grüss-Voronovskaya type theorem. In the last section, we estimate the rates of pointwise approximation of Kantorovich variant of Chlodowsky Szász operators induced by Boas-Buck type polynomials for functions with derivatives of bounded variation.
In this paper, we introduce and investigate a new class of the parametric generalization of the B... more In this paper, we introduce and investigate a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators, which considerably extends the well-known class of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For this new class of approximation operators, we present a Korovkin type theorem and a Grüss-Voronovskaya type theorem, and also study the rate of its convergence. Moreover, we derive several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we prove some shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and, as a special case, we deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators.
We enumerate partitions of the set $\{1,\dots,n\}$ according to occurrences of isolated successio... more We enumerate partitions of the set $\{1,\dots,n\}$ according to occurrences of isolated successions, that is, integer strings $a,a+1,\dots,b$ in a block when neither $a-1$ nor $b+1$ lies in the same block. Our results include explicit formulas and generating functions for the number of partitions containing isolated successions of a given length. We also consider a corresponding analog of the associated Stirling numbers of the second kind.
We consider various statistics on the set Fn consisting of the distinct permutations of length n+... more We consider various statistics on the set Fn consisting of the distinct permutations of length n+1 that arise as a flattening of some partition of the same size. In particular, we enumerate members of Fn according to the number of occurrences of three-letter consecutive patterns, considered more broadly in the context of r-partitions. As special cases of our results, we obtain formulas for the number of members of Fn avoiding a given consecutive pattern and for the total number of occurrences of a pattern over all members of Fn.
Abstract. Catalan words are a particular case of growth-restricted words. Here we give the bivari... more Abstract. Catalan words are a particular case of growth-restricted words. Here we give the bivariate generating function for bargraphs associated to Catalan words according to the semiperimeter and area statistics. Exact formulas to count Catalan bargraphs according to the two statistics are also found. We show a connection to the Narayana numbers. Finally, we give similar results for the exterior corner statistic.
Discrete Mathematics, Algorithms and Applications, 2021
In this paper, we introduce a new statistic on standard Young tableaux that is closely related to... more In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having [Formula: see text] cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.
An inversion sequence of length $n$ is a sequence of integers $e=e_1\cdots e_n$ which satisfies ... more An inversion sequence of length $n$ is a sequence of integers $e=e_1\cdots e_n$ which satisfies for each $i\in[n]=\{1,2,\ldots,n\}$ the inequality $0\le e_i < i$. For a set of patterns $P$, we let $\mathbf{I}_n(P)$ denote the set of inversion sequences of length $n$ that avoid all the patterns from~$P$. We say that two sets of patterns $P$ and $Q$ are I-Wilf-equivalent if $|\mathbf{I}_n(P)|=|\mathbf{I}_n(Q)|$ for every~$n$. In this paper, we show that the number of I-Wilf-equivalence classes among triples of length-3 patterns is $137$, $138$ or~$139$. In particular, to show that this number is exactly $137$, it remains to prove $\{101,102,110\}\stackrel{\mathbf{I}}{\sim}\{021,100,101\}$ and $\{100,110,201\}\stackrel{\mathbf{I}}{\sim}\{100,120,210\}$.
Discrete Mathematics & Theoretical Computer Science
Combinatorics We consider the sum u of differences between adjacent letters of a word of n letter... more Combinatorics We consider the sum u of differences between adjacent letters of a word of n letters, chosen uniformly at random from a given alphabet. This paper obtains the enumerating generating function for the number of such words with respect to the sum u, as well as explicit formulas for the mean and variance of u.
Discrete Mathematics & Theoretical Computer Science
Let asc and desc denote respectively the statistics recording the number of ascents or descents i... more Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and ...
In this paper we introduce the concept of s-Fuss-Catalan words. This new family of words generali... more In this paper we introduce the concept of s-Fuss-Catalan words. This new family of words generalizes the Catalan words (taking s = 1), which are a particular case of growth-restricted words. Here we enumerate the polyominoes or bargraphs associated with the s-Fuss-Catalan words according to the semiperimeter and area statistics. Additionally, we obtain combinatorial formulas to count the s-Fuss-Catalan bargraphs according of these statistics.
We introduce a decomposition method for column-convex polyominoes and enumerate them in terms of ... more We introduce a decomposition method for column-convex polyominoes and enumerate them in terms of two statistics: the number of internal vertices and the number of corners in the boundary. We first find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of interior vertices. In particular, we show that the average number of interior vertices over all column-convex polyominoes of perimeter 2 n is asymptotic to $$\alpha _o n^{3/2}$$ α o n 3 / 2 where $$\alpha _o\approx 0.57895563\ldots .$$ α o ≈ 0.57895563 … . We also find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of corners in the boundary. In particular, we show that the average number of corners over all column-convex polyominoes of perimeter 2 n is asymptotic to $$\alpha _1n$$ α 1 n where $$\alpha _1\approx 1.17157287\ldots .$$ α 1 ≈ 1.17157287 … .
In this paper, we study the Kantorovich variant of Chlodowsky Szász operators induced by Boas-Buc... more In this paper, we study the Kantorovich variant of Chlodowsky Szász operators induced by Boas-Buck type polynomials and prove a Korovkin type theorem. Also, we present the Voronovskaya type theorem and Grüss-Voronovskaya type theorem. In the last section, we estimate the rates of pointwise approximation of Kantorovich variant of Chlodowsky Szász operators induced by Boas-Buck type polynomials for functions with derivatives of bounded variation.
In this paper, we introduce and investigate a new class of the parametric generalization of the B... more In this paper, we introduce and investigate a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators, which considerably extends the well-known class of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For this new class of approximation operators, we present a Korovkin type theorem and a Grüss-Voronovskaya type theorem, and also study the rate of its convergence. Moreover, we derive several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we prove some shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and, as a special case, we deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators.
We enumerate partitions of the set $\{1,\dots,n\}$ according to occurrences of isolated successio... more We enumerate partitions of the set $\{1,\dots,n\}$ according to occurrences of isolated successions, that is, integer strings $a,a+1,\dots,b$ in a block when neither $a-1$ nor $b+1$ lies in the same block. Our results include explicit formulas and generating functions for the number of partitions containing isolated successions of a given length. We also consider a corresponding analog of the associated Stirling numbers of the second kind.
We consider various statistics on the set Fn consisting of the distinct permutations of length n+... more We consider various statistics on the set Fn consisting of the distinct permutations of length n+1 that arise as a flattening of some partition of the same size. In particular, we enumerate members of Fn according to the number of occurrences of three-letter consecutive patterns, considered more broadly in the context of r-partitions. As special cases of our results, we obtain formulas for the number of members of Fn avoiding a given consecutive pattern and for the total number of occurrences of a pattern over all members of Fn.
Abstract. Catalan words are a particular case of growth-restricted words. Here we give the bivari... more Abstract. Catalan words are a particular case of growth-restricted words. Here we give the bivariate generating function for bargraphs associated to Catalan words according to the semiperimeter and area statistics. Exact formulas to count Catalan bargraphs according to the two statistics are also found. We show a connection to the Narayana numbers. Finally, we give similar results for the exterior corner statistic.
Discrete Mathematics, Algorithms and Applications, 2021
In this paper, we introduce a new statistic on standard Young tableaux that is closely related to... more In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having [Formula: see text] cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.
This book gives an introduction to and an overview of the methods used in the combinatorics of pa... more This book gives an introduction to and an overview of the methods used in the combinatorics of pattern avoidance and pattern enumeration in set partitions, a very active area of research in the last decade. The first known application of set partitions arose in the context of tea ceremonies and incense games in Japanese upper class society around A.D. 1500. Guests at a Kado ceremony would be smelling cups with burned incense with the goal to either identity the incense or to identity which cups contained identical incense. There are many variations of the game, even today. One particular game is named genji-ko, and it is the one that originated the interest in set partitions. Five different incense were cut into five pieces, each piece put into a separate bag, and then five of these bags were chosen to be burned. Guests had to identify which of the five incense were the same. The Kado ceremony masters developed symbols for the different possibilities, so-called genji-mon. Each such symbol consists of five vertical bars, some of which are connected by horizontal bars. Fifty-two symbols were created, and for easier memorization, each symbol was identified with one of the fifty-four chapters of the famous \emph{Tale of Genji} by Lady Murasaki. In time, these genji-mon and two additional symbols started to be displayed at the beginning of each chapter of the \emph{Tale of Genji} and in turn became part of numerous Japanese paintings. They continued to be popular symbols for family crests and Japanese kimono patterns in the early 20th century and can be found on T-shirts sold today.
Until the late 1960s, individual research papers on various aspects of set partitions appeared, but there was no focused research interest. This changed after the 1970s, when several groups of authors developed new research directions. They studied set partitions under certain set of conditions, and not only enumerated the total number of these objects, but also certain of their characteristics. Other focuses were the relation between the algebra and set partitions such as the noncrossing set partitions, the appearance of set partitions in physics where the number of set partitions is a good language to find explicit formula for normal ordering form of an expression of boson operators, and the study of random set partition to obtain asymptotics results.
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We wrote this book to provide a comprehensive resource for anybody interested in this new area of research. It will combine the following in one place:
(1) provide a self-contained, broadly accessible introduction to research in this area,
(2) present an overview of the history of research on enumeration of and pattern avoidance in set partitions,
(3) present several links between set partitions and other areas of mathematics, and in normal ordering form of expressions of boson operators in physics,
(4) describe a variety of tools and approaches that are also useful to other areas of enumerative combinatorics.
(5) suggest open questions for further research, and
(6) provide a comprehensive and extensive bibliography.
Our book is based on my own research and on that of my collaborators and other researchers in the field. We present these results with consistent notation and have modified some proofs to relate to other results in the book. As a general rule, theorems listed without specific references give results from articles by the author and their collaborators, while results from other authors are given with specific references. Many results of my own, with or without collaborators, articles are omitted, so we refer the reader to all these articles to find the full details in the subject.
This text gives an introduction to and an overview of the methods used in the combinatorics of pa... more This text gives an introduction to and an overview of the methods used in the combinatorics of pattern avoidance and pattern enumeration in compositions and words, a very active area of research in the last decade. The earliest results on enumeration of compositions appeared in a paper by P.A. MacMahon in 1893, while Axel Thue is credited with starting research on the combinatorics of words in a paper in 1906. MacMahon considered words in the context of partitions and permutations, while Thue approached words from a number theoretic background. In the decades that followed the main focus of research in
enumerative combinatorics was on partitions and permutations, driven in part by MacMahon's prolific writing. Until the late 1960s,
individual articles on various aspects of compositions appeared, but there was no concentrated research interest.
This changed in the 1970s, when several groups of authors developed new research directions. They studied compositions and words that were restricted in some way, and not only enumerated the total number of these objects, but also certain of their characteristics (statistics). Another focus was the study of random compositions to obtain asymptotic results. Most publications on compositions and words have been within the last decade. Authors have studied various aspects of compositions and words, generalizing previous results and introducing new concepts. In particular, research on pattern avoidance in words and compositions has followed earlier very active research on pattern avoidance in permutations.
Goals:
We see how partial genus distributions behave under bar-path amalgamation. We give expli... more Goals:
We see how partial genus distributions behave under bar-path amalgamation. We give explicit formulas for the pgd's of bar-path using a repeated base graph $(G,u,v)$ and the genus distribution when we complete the bar-path with a self-bar to form a bar-ring.
We apply these formulas to six different base graphs $(G,u,v)$ with LC genus distributions and LC pgd's, and we show that all the pgd's for the bar-path are LC and that the genus distribution for the bar-ring is LC.
You are welcome to subscribe to ECA channel that we opened regarding the journal and the conferen... more You are welcome to subscribe to ECA channel that we opened regarding the journal and the conference, where you can find all the lectures that are given at the conference several days ago. Just click on the following link and subscribe https://www.youtube.com/channel/UCeoIJ1uLMqQHwybt4SFreOw/featured Please encourage your friends too. It is helpful for ICECA2023.
Uploads
Papers by Toufik Mansour
Until the late 1960s, individual research papers on various aspects of set partitions appeared, but there was no focused research interest. This changed after the 1970s, when several groups of authors developed new research directions. They studied set partitions under certain set of conditions, and not only enumerated the total number of these objects, but also certain of their characteristics. Other focuses were the relation between the algebra and set partitions such as the noncrossing set partitions, the appearance of set partitions in physics where the number of set partitions is a good language to find explicit formula for normal ordering form of an expression of boson operators, and the study of random set partition to obtain asymptotics results.
%----
We wrote this book to provide a comprehensive resource for anybody interested in this new area of research. It will combine the following in one place:
(1) provide a self-contained, broadly accessible introduction to research in this area,
(2) present an overview of the history of research on enumeration of and pattern avoidance in set partitions,
(3) present several links between set partitions and other areas of mathematics, and in normal ordering form of expressions of boson operators in physics,
(4) describe a variety of tools and approaches that are also useful to other areas of enumerative combinatorics.
(5) suggest open questions for further research, and
(6) provide a comprehensive and extensive bibliography.
Our book is based on my own research and on that of my collaborators and other researchers in the field. We present these results with consistent notation and have modified some proofs to relate to other results in the book. As a general rule, theorems listed without specific references give results from articles by the author and their collaborators, while results from other authors are given with specific references. Many results of my own, with or without collaborators, articles are omitted, so we refer the reader to all these articles to find the full details in the subject.
enumerative combinatorics was on partitions and permutations, driven in part by MacMahon's prolific writing. Until the late 1960s,
individual articles on various aspects of compositions appeared, but there was no concentrated research interest.
This changed in the 1970s, when several groups of authors developed new research directions. They studied compositions and words that were restricted in some way, and not only enumerated the total number of these objects, but also certain of their characteristics (statistics). Another focus was the study of random compositions to obtain asymptotic results. Most publications on compositions and words have been within the last decade. Authors have studied various aspects of compositions and words, generalizing previous results and introducing new concepts. In particular, research on pattern avoidance in words and compositions has followed earlier very active research on pattern avoidance in permutations.
We see how partial genus distributions behave under bar-path amalgamation. We give explicit formulas for the pgd's of bar-path using a repeated base graph $(G,u,v)$ and the genus distribution when we complete the bar-path with a self-bar to form a bar-ring.
We apply these formulas to six different base graphs $(G,u,v)$ with LC genus distributions and LC pgd's, and we show that all the pgd's for the bar-path are LC and that the genus distribution for the bar-ring is LC.
Please encourage your friends too. It is helpful for ICECA2023.